The Annals of Applied Statistics

Functional factor analysis for periodic remote sensing data

Chong Liu, Surajit Ray, Giles Hooker, and Mark Friedl

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We present a new approach to factor rotation for functional data. This is achieved by rotating the functional principal components toward a predefined space of periodic functions designed to decompose the total variation into components that are nearly-periodic and nearly-aperiodic with a predefined period. We show that the factor rotation can be obtained by calculation of canonical correlations between appropriate spaces which make the methodology computationally efficient. Moreover, we demonstrate that our proposed rotations provide stable and interpretable results in the presence of highly complex covariance. This work is motivated by the goal of finding interpretable sources of variability in gridded time series of vegetation index measurements obtained from remote sensing, and we demonstrate our methodology through an application of factor rotation of this data.

Article information

Ann. Appl. Stat. Volume 6, Number 2 (2012), 601-624.

First available in Project Euclid: 11 June 2012

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

Mathematical Reviews number (MathSciNet)


Liu, Chong; Ray, Surajit; Hooker, Giles; Friedl, Mark. Functional factor analysis for periodic remote sensing data. Ann. Appl. Stat. 6 (2012), no. 2, 601--624. doi:10.1214/11-AOAS518.

Export citation


  • Allen, G. I., Grosenick, L. and Taylor, J. (2011). A generalized least squares matrix decomposition. Technical report, Rice Univ.
  • Everson, R., Cornillon, P., Sirovich, L. and Webber, A. (1996). Empirical eigenfunction analysis of sea surface temperatures in the Western North Atlantic. AIP Conf. Proc. 375 563–590.
  • Gervini, D. and Gasser, T. (2004). Self-modelling warping functions. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 959–971.
  • Hall, P., Müller, H.-G. and Wang, J.-L. (2006). Properties of principal component methods for functional and longitudinal data analysis. Ann. Statist. 34 1493–1517.
  • Hartmann, D. L. (1994). Global Physical Climatology. Academic Press, New York.
  • He, G., Müller, H.-G. and Wang, J.-L. (2003). Functional canonical analysis for square integrable stochastic processes. J. Multivariate Anal. 85 54–77.
  • Holton, J. (1992). An Introduction to Dynamic Meteorology. International Geophysics Series. Academic Press, San Diego, New York.
  • Huete, A., Didan, K., Miura, T., Rodriguez, E. P., Gao, X. and Ferreira, L. G. (2002). Overview of the radiometric and biophysical performance of the MODIS vegetation indices. Remote Sensing of Environment 83 195–213.
  • Kneip, A. and Ramsay, J. O. (2008). Combining registration and fitting for functional models. J. Amer. Statist. Assoc. 20 1266–1305.
  • Koulis, T., Ramsay, J. O. and Levitin, D. J. (2008). From zero to sixty: Calibrating real-time responses. Psychometrika 73 321–339.
  • Leurgans, S. E., Moyeed, R. A. and Silverman, B. W. (1993). Canonical correlation analysis when the data are curves. J. Roy. Statist. Soc. Ser. B 55 725–740.
  • Li, P.-L. and Chiou, J.-M. (2011). Identifying cluster number for subspace projected functional data clustering. Comput. Statist. Data Anal. 55 2090–2103.
  • Li, Y. and Hsing, T. (2010). Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data. Ann. Statist. 38 3321–3351.
  • Liu, X. and Müller, H.-G. (2004). Functional convex averaging and synchronization for time-warped random curves. J. Amer. Statist. Assoc. 99 687–699.
  • Liu, C., Ray, S., Hooker, G. and Friedl, M. (2012). Supplement to “Functional factor analysis for periodic remote sensing data.” DOI:10.1214/11-AOAS518SUPP.
  • Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.
  • Müller, H.-G., Stadtmüller, U. and Yao, F. (2006). Functional variance processes. J. Amer. Statist. Assoc. 101 1007–1018.
  • Parmesan, C. and Yohe, G. (2003). A globally coherent fingerprint of climate change impacts across natural systems. Nature 421 37–42.
  • Peng, J. and Paul, D. (2009). A geometric approach to maximum likelihood estimation of the functional principal components from sparse longitudinal data. J. Comput. Graph. Statist. 18 995–1015.
  • Piao, S. L., Ciais, P., Friedlingstein, P., Peylin, P., Reichstein, M., Luyssaert, S., Margolis, H., Fang, J. Y., Barr, A., Chen, A. P., Grelle, A., Hollinger, D. Y., Laurila, T., Lindroth, A., Richardson, A. D. and Vesala, T. (2008). Net carbon dioxide losses of northern ecosystems in response to autumn warming. Nature 451 49–52.
  • R Development Core Team. (2010). R: A Language and Environment for Statistical Computing. Vienna, Austria ISBN 3-900051-07-0.
  • Ramsay, J. O. and Silverman, B. W. (2002). Applied Functional Data Analysis. Springer, New York.
  • Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis, 2nd ed. Springer, New York.
  • Ramsay, J. O., Wickham, H., Graves, S. and Hooker, G. (2010). fda: Functional data analysis. R package version 2.2.2.
  • Silverman, B. W. (1996). Smoothed functional principal components analysis by choice of norm. Ann. Statist. 24 1–24.
  • Yao, F., Müller, H.-G. and Wang, J.-L. (2005). Functional linear regression analysis for longitudinal data. Ann. Statist. 33 2873–2903.
  • Zhang, X., Friedl, M. A. and Schaaf, C. B. (2006). Global vegetation phenology from moderate resolution imaging spectroradiometer (MODIS): Evaluation of global patterns and comparison with in situ measurements. Journal of Geophysical Research 111 G04017.

Supplemental materials

  • Supplementary material: Description of data and details of simulation. The supplementary material is divided into 3 sections. The first section provides a detailed description of the Harvard Forest data that is used in this article, including preprocessing steps. We also provide a detailed description of the imputation steps for pixels with missing observations. The second section provides a description of Annual Information and its application is demonstrated through a simulation study. The last section provides results related to the bootstrap hypothesis testing procedure proposed in this article. In particular, we present the test results on the Harvard Forest data and simulation studies where we explore the empirical power curve and size on simulated data sets.